Investment Timing Option: Option Analysis

The Karns Oil Company is deciding whether to drill for oil on a tract of land that the company owns. The company estimates the project would cost $$8$ million today. Karns estimates that, once drilled, the oil will generate positive net cash flows of $$4$ million a year at the end of each of the next $4$ years. Although the company is fairly confident about its cash flow forecast, in $2$ years it will have more information about the local geology and about the price of oil. Karns estimates that if it waits $2$ years then the project would cost $$9$ million. Moreover, if it waits $2$ years, then there is a $90\%$ chance that the net cash flows would be $$4\mathrm{.}2$ million a year for $4$ years and a $10\%$ chance that they would be $$2\mathrm{.}2$ million a year for $4$ years. Assume all cash flows are discounted at $10\%.$ Use the Black$-$Scholes model to estimate the value of the option. Assume the variance of the project's rate of return is $0\mathrm{.}512$ and that the risk$-$free rate is $6\%.$ Do not round intermediate calculations. Enter your answer in millions. For example, an answer of $$1\mathrm{.}234$ million should be entered as $1\mathrm{.}234,$ not $1,234,000.$ Round your answer to three decimal places.